1. Reference problem

1.1. Theoretical framework

The unknowns of the problem are the degrees of freedom of movement and nodal damage. It is then a question of minimizing an energy of the form:

\(\phi (u,\alpha )\mathrm{=}\frac{1}{2}A(d)E{\epsilon }^{2}+\psi (d)+\frac{c}{2}\mathrm{\nabla }\alpha \mathrm{.}\mathrm{\nabla }\alpha\)

Where \(E\) is the Young’s modulus of the material, \(A(d)\) the stiffness function, the stiffness function, \(\psi (d)\) the dissipation, and \(c\) the non-local coefficient.

In the case of law ENDO_CARRE:

\(A(d)={(1-d)}^{2}\) and \(\psi (d)=\frac{{\sigma }_{y}^{2}}{E}d\)

The criterion corresponding to law ENDO_CARRE, for a homogeneous solution (\(\nabla \alpha =0\)), is therefore written:

\(d=1-(\frac{{W}_{y}}{{W}_{\mathrm{el}}})\)

Where \({W}_{\mathrm{el}}\) is the elastic deformation energy and:

\({W}_{y}=\frac{{\sigma }_{y}^{2}}{2E}\)

1.2. Geometry

Consider a square with side \(L=1\text{m}\).

Dx

Figure 1: Representation of the problem

1.3. Material properties

1.3.1. Law of damage: material ENDO_CARRE

Elastic characteristics:

\(E=1\text{Pa}\)

\(\nu =0.\)

Characteristics related to the law of damage:

Elasticity limit:

\(\mathrm{SY}=0.01\text{Pa}\)

Non-local characteristics:

\(c=1.0\text{N}\)

1.4. Boundary conditions and loads

Embed: Imposed movements zero DY = 0 \(m\) on the bottom and top horizontal edges (\(y=0.\) and \(y=1.\)) and DX = 0 \(m\) on the left edge (\(x=0.\)). See Figure 1.

Loading 1: Imposed displacement \({U}_{1}\) on the right vertical edge:

At moment \({t}_{1}\): DX = \(0.01\text{m}\)

At moment \({t}_{2}\): DX = \(0.0125\text{m}\)

At moment \({t}_{3}\): DX = \(0.02\text{m}\)